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this sequence was given as a practice problem and I'm really having trouble. Heres the question:

let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f $ uniformly on $[0, \infty)$

My Work: I determined that the pointwise limit $f(x) = $lim$f_n(x)$ has to be $1$ but I am not too sure what to do after this. Any help is appreciated

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Is the pointwise limit a continuous function? Each function is continuous, so if convergence were uniform, the limit would be too.

Ans No, it is not. It is $0$ in $[0,1)$, $1/2$ at $1$ and $1$ over $(1,\infty]$.

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  • $\begingroup$ the question doesnt specify. All that it says is that it is a sequence $\endgroup$ – user3182418 Apr 8 '14 at 3:48
  • $\begingroup$ @user3182418 I am asking you. Is the pointwise limit a continuous function? $\endgroup$ – Pedro Tamaroff Apr 8 '14 at 3:48
  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. $\endgroup$ – user127096 Apr 8 '14 at 4:07
  • $\begingroup$ @cheapeffectivedietpills It does. $\endgroup$ – Pedro Tamaroff Apr 8 '14 at 4:10
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you can use the following characterization : $f_n $ converges uniformly to $f$ on $S$ iff

$$ \lim_{n \to \infty} \sup_{x \in S} \{ |f_n(x) - f(x) | \} = 0$$

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